Dedekind cuts (which are necessary for epsilon-delta proofs) or alternatively Bolzano Weierstrass theorem (if you use sequential convergence definition) imply m

Dedekind cuts (which are necessary for epsilon-delta proofs) or alternatively Bolzano Weierstrass theorem (if you use sequential convergence definition) imply multiple sizes of infinity, so the logical arrow is in the other direction, roughly: "real analysis implies assigning different sizes to sets", not "assigning different sizes to sets implies real analysis". It's been a while but iirc there are other constructions (such as p-adic numbers) which create multiple sizes of infinities out of the rationals which aren't real analysis.